How to disable only a visual in renders and not the effect
Of course there is a way.

You need to make your fish image transparent.
Not the whole image, just the grey parts.

Here is how you do it:
01) Use Photoshop, Gimp or other software to remove the fish image background.
02) Save it as a PNG.
03) Add it to Blender and set the nodes like in my example.  

Remove image background


Enable transparency for your image